Import Question JSON

Current Question (ID: 8831)

Question:

$\text{A particle moves along a straight line and its position as a function of time is given by } x = t^3 - 3t^2 + 3t + 3 \text{ then particle:}$

Options:

1. $\text{stops at } t = 1 \text{ s and reverses its direction of motion.}$
2. $\text{stops at } t = 1 \text{ s and continues further without a change of direction.}$
3. $\text{stops at } t = 2 \text{ s and reverses its direction of motion.}$
4. $\text{stops at } t = 2 \text{ s and continues further without a change of direction.}$

Solution:

$\text{Hint: Check the sign of velocity.}$ $\text{Step: Find the motion of the particle.}$ $x = t^3 - 3t^2 + 3t + 3$ $\Rightarrow v = \frac{dx}{dt} = 3t^2 - 6t + 3$ $\text{Put, } v = 0$ $\Rightarrow 3t^2 - 6t + 3 = 0$ $\Rightarrow t^2 - 2t + 1 = 0$ $\Rightarrow (t - 1)^2 = 0 \Rightarrow t = 1 \text{ s}$ $\text{So, the particle stops only once at } t = 1 \text{ s.}$ $\Rightarrow v = 3(t - 1)^2 > 0$ $\text{Therefore, the particle will not reverse the direction and continue further without a change of direction.}$ $\text{Hence, option (2) is the correct answer.}$

Import JSON File

Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}